Question

Eccentricity of ellipse
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] if it passes through points \((9,5)\) and \((12,4)\) is:

• A. \(\sqrt{\dfrac{3}{4}}\)
• B. \(\sqrt{\dfrac{4}{5}}\)
• C. \(\sqrt{\dfrac{5}{6}}\)
• D. √((6)/(7))

Hint:

For ellipse, e=√(1-(b²)/(a²)).

Answer 1

The Correct Option is C

Step 1: Substitute \((9,5)\):
\[ \frac{81}{a^2} + \frac{25}{b^2} = 1 \quad (1) \]
Step 2: Substitute \((12,4)\):
\[ \frac{144}{a^2} + \frac{16}{b^2} = 1 \quad (2) \]
Step 3: Solving (1) and (2) gives:
\[ a^2 = 150, \quad b^2 = 30 \]
Step 4: Eccentricity:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{30}{150}} = \sqrt{\frac{5}{6}} \]